{"id":2099,"date":"2015-11-12T18:30:42","date_gmt":"2015-11-12T18:30:42","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=2099"},"modified":"2015-11-12T18:30:42","modified_gmt":"2015-11-12T18:30:42","slug":"finding-the-number-of-permutations-of-n-non-distinct-objects","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/finding-the-number-of-permutations-of-n-non-distinct-objects\/","title":{"raw":"Finding the Number of Permutations of n Non-Distinct Objects","rendered":"Finding the Number of Permutations of n Non-Distinct Objects"},"content":{"raw":"

We have studied permutations where all of the objects involved were distinct. What happens if some of the objects are indistinguishable? For example, suppose there is a sheet of 12 stickers. If all of the stickers were distinct, there would be [latex]12![\/latex] ways to order the stickers. However, 4 of the stickers are identical stars, and 3 are identical moons. Because all of the objects are not distinct, many of the [latex]12![\/latex] permutations we counted are duplicates. The general formula for this situation is as follows.\n<\/p>

[latex]\\frac{n!}{{r}_{1}!{r}_{2}!\\dots {r}_{k}!}[\/latex]<\/div>\nIn this example, we need to divide by the number of ways to order the 4 stars and the ways to order the 3 moons to find the number of unique permutations of the stickers. There are [latex]4![\/latex] ways to order the stars and [latex]3![\/latex] ways to order the moon.\n
[latex]\\frac{12!}{4!3!}=3\\text{,}326\\text{,}400[\/latex]<\/div>\nThere are 3,326,400 ways to order the sheet of stickers.\n
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A General Note: Formula for Finding the Number of Permutations of n<\/em> Non-Distinct Objects<\/h3>\nIf there are [latex]n[\/latex] elements in a set and [latex]{r}_{1}[\/latex] are alike, [latex]{r}_{2}[\/latex] are alike, [latex]{r}_{3}[\/latex] are alike, and so on through [latex]{r}_{k}[\/latex], the number of permutations can be found by\n
[latex]\\frac{n!}{{r}_{1}!{r}_{2}!\\dots {r}_{k}!}[\/latex]<\/div>\n<\/div>\n
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Example 6: Finding the Number of Permutations of n<\/em> Non-Distinct Objects<\/h3>\nFind the number of rearrangements of the letters in the word DISTINCT.\n\n<\/div>\n
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Solution<\/h3>\nThere are 8 letters. Both I and T are repeated 2 times. Substitute [latex]n=8, {r}_{1}=2, [\/latex] and [latex] {r}_{2}=2 [\/latex] into the formula.\n
[latex]\\frac{8!}{2!2!}=10\\text{,}080 [\/latex]<\/div>\nThere are 10,080 arrangements.\n\n<\/div>\n
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Try It 10<\/h3>\nFind the number of rearrangements of the letters in the word CARRIER.\n\nSolution<\/a>\n\n<\/div>","rendered":"

We have studied permutations where all of the objects involved were distinct. What happens if some of the objects are indistinguishable? For example, suppose there is a sheet of 12 stickers. If all of the stickers were distinct, there would be [latex]12![\/latex] ways to order the stickers. However, 4 of the stickers are identical stars, and 3 are identical moons. Because all of the objects are not distinct, many of the [latex]12![\/latex] permutations we counted are duplicates. The general formula for this situation is as follows.\n<\/p>\n

[latex]\\frac{n!}{{r}_{1}!{r}_{2}!\\dots {r}_{k}!}[\/latex]<\/div>\n

In this example, we need to divide by the number of ways to order the 4 stars and the ways to order the 3 moons to find the number of unique permutations of the stickers. There are [latex]4![\/latex] ways to order the stars and [latex]3![\/latex] ways to order the moon.<\/p>\n

[latex]\\frac{12!}{4!3!}=3\\text{,}326\\text{,}400[\/latex]<\/div>\n

There are 3,326,400 ways to order the sheet of stickers.<\/p>\n

\n

A General Note: Formula for Finding the Number of Permutations of n<\/em> Non-Distinct Objects<\/h3>\n

If there are [latex]n[\/latex] elements in a set and [latex]{r}_{1}[\/latex] are alike, [latex]{r}_{2}[\/latex] are alike, [latex]{r}_{3}[\/latex] are alike, and so on through [latex]{r}_{k}[\/latex], the number of permutations can be found by<\/p>\n

[latex]\\frac{n!}{{r}_{1}!{r}_{2}!\\dots {r}_{k}!}[\/latex]<\/div>\n<\/div>\n
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Example 6: Finding the Number of Permutations of n<\/em> Non-Distinct Objects<\/h3>\n

Find the number of rearrangements of the letters in the word DISTINCT.<\/p>\n<\/div>\n

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Solution<\/h3>\n

There are 8 letters. Both I and T are repeated 2 times. Substitute [latex]n=8, {r}_{1}=2,[\/latex] and [latex]{r}_{2}=2[\/latex] into the formula.<\/p>\n

[latex]\\frac{8!}{2!2!}=10\\text{,}080[\/latex]<\/div>\n

There are 10,080 arrangements.<\/p>\n<\/div>\n

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Try It 10<\/h3>\n

Find the number of rearrangements of the letters in the word CARRIER.<\/p>\n

Solution<\/a><\/p>\n<\/div>\n\n\t\t\t

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